Sismik Prospeksiyon ( ingilizce ) Introduction to Seismology - 10 Introduction to Seismology: Lecture Notes on 17 th and 19 th March, 2008 3)Fr ee Surface (P-SV sys tem) We have This is the Critical Condition. . So, as we increase ( , we have a situation when . Sub Critical: Critical for P: Post Critical : ; ; Now it is critical for P We can have a case when . This gives rise to Evanescent P,S waves which couple to give LOVE Waves. We shall learn about them in coming lectures. Reflection and Transmi s sion Coeff i cien ts f or SH Wave interacti ng at a W e lded In terface: Define Potential s : Consider an SH-wave incident at an interface between two layers of different elastic properties. Because SH waves are decoupled from P and SV waves, the transmitted and reflected waves will also be SH waves. Recall that displacement corresponding to SH waves ( ) satisfies the wave equation. So, SH part EZ Ğ ?Y?l ? Ğ ??s? Ğ D Z l Ğ Yls Au ? l Z IsY E ? Zu ?ls Z Y?? ?l ? Ğ D Z ? uE A Y Au ?? Ğ ^, D A ?l l S ? Z ? OS D Z l Ğ Yls Au ? l ZZ ? & Z ? simplicity, we deal with displacements directly instead of potentials. Introduction to Seismology: Lecture Notes on 17 th and 19 th March, 2008 Complied by: Sudhish Kumar Bakku The displacement of the down going wave in the Layer1 can be expressed as Where for notational convenience And the reflected, up going wave in Layer1 can be expressed as Now, setting the interface to z =0, the total displacement in Layer1 is The displacement in Layer2 is given by Appl y i n g B o undar y C o ndi t i ons : The kinematic boundary conditions require that the displacement be continuous across the interface at z = 0. Therefore, setting z = 0 and equating the displacements in Layers 1 and 2, we arrive at the following statement: The dynamic boundary conditions require that tractions are continuous. We have, And recalling, for SH waves, , we have: Introduction to Seismology: Lecture Notes on 17 th and 19 th March, 2008 Complied by: Sudhish Kumar Bakku hence, @z=0, The preceding facts imply that Fo rmulatin g Zoe pprit z Equ a tions: The reflection and transmission coefficients are expressed as If we arrange Eqn 2 Combining eqns 1 and 2 give the Zo e p p ritz equation Introduction to Seismology: Lecture Notes on 17 th and 19 th March, 2008 Complied by: Sudhish Kumar Bakku Where the left hand side is the incoming wave and the RHS are the resulting waves. S o lving the Equations for Reflect i on an d T r ans m iss i on Coefficien ts: Again, if we arrange the Zo e p p ritz equation: Then the reflection coefficient is And to calculate the transmission coefficient: The dependence on the incidence angle is illustrated by And Introduction to Seismology: Lecture Notes on 17 th and 19 th March, 2008 Complied by: Sudhish Kumar Bakku Using these variables to rewrite the reflection and transmission coefficients: The reflection and transmission coefficients are equal to the RHS of the equations above for the specific case of normal incidence, when For a welded surface we can also have Thus we can define a scatter matrix S Behaviour of Refle ction an d T r ans m is sion Coefficien t s with an g l e of in cidence: We have, As we have discussed before, since and are functions of ray parameter, R and T depend upon the incident angle. Introduction to Seismology: Lecture Notes on 17 th and 19 th March, 2008 Complied by: Sudhish Kumar Bakku Say is the incident angle for which . So, is the critical angle Pre-Cr i ti cal: . Now, we can express as below: Substituting the above in the relation for R and T, we have: Let ? s assume that . So, when , we have R<0 and 0
0 and T >1 until we reach the critical angle. Cri ti cal Angl e: If we increase the angle such that and we have . Thus, R = 1 and T=2. Introduction to Seismology: Lecture Notes on 17 th and 19 th March, 2008 Complied by: Sudhish Kumar Bakku P o s t - C ri ti c a l A n g l e : If we increase the angle beyond , can ? t further change and remains at . Now we have: For the incoming wave . There fore, is now a Complex Number. Since, is complex, we have both R and T to be Complex. R and T are plotted as a function of the incidence angle (see figure below). The real part is in thin solid line, imaginary part in dashed line and absolute value in bold line. Introduction to Seismology: Lecture Notes on 17 th and 19 th March, 2008 Complied by: Sudhish Kumar Bakku Incidence angle in Degrees Incidence angle in Degrees Consider the transmitted wave: For a SH wave, the displacement vector is In a post critical situation when , Where the first term on the RHS describes an exponential decay in the z-direction (there is no propagation in the z-direction). The frequency, ? , controls the rate of the decay. The second term on the RHS describes a harmonic function ( x ,t), therefore the wave propagation is in the x-direction. The property of deceasing wave amplitude with depth based on the frequency of the wave is known as the Evanescence. Introduction to Seismology: Lecture Notes on 17 th and 19 th March, 2008 Complied by: Sudhish Kumar Bakku A wave with a short wavelength ( ? ), high frequency ( ? ), will decay more quickly. At infinite frequency the decay is instantaneous and the wave becomes a ray. If the properties of a medium change with depth, for example there is a body which allows a wave to pass through it more quickly at depth only the low ? , long ? , waves will sample it as the high waves will have been stripped out. A wave with frequency dependence is a D i sp e r si ve wa ve . Often a wave can be dispersive and evanescent. Consider the Reflected Wave: The reflection coefficient is Where, Now, if we write the displacements of the reflected wave, Where, is the amplitude of the incident wave. Thus, we can observe that the reflected wave has a phase shift of . For a given incident angle, as we observe, different frequencies have different phase shifts. So, a single incident spike would be recorded as a spread waveform at the receiver as different frequencies are shifted apart in phase. References and Sources of figures: Introduction to Seismology: Stein and Wysesion Class Notes on 2/23/05: Compiled by Sophie Michelet Class Notes on 2/28/05 Class Notes on 3/2/05 Introduction to Seismology: Lecture Notes on 17 th and 19 th March, 2008 Complied by: Sudhish Kumar Bakku